Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

418 MR BAKER, ON THE THEORY OF FUNCTIONS
taken over a closed (n-2)-fold, which (see ~ 9 below) may be interpreted as
(el)/(1 2)(x13 + it14 + iK23- t24) dSn-2,
is equal to the integral
f L( + iK2) ( + i -( / + i(4) \ + 0 72) J
taken over the (n-1)-fold bounded by the (n-2)-fold; and this vanishes identically on
account of
(a t 2 z a (3O, +%i^ f = 0.
It is supposed that the original (n - 2)-fold of integration is not one given by the vanishing
of a single equation involving the two complex variables, since otherwise (cf. ~ 9 below)
K13 = I24, K^14= - C23, and therefore
i13 + iKl4 + iK23 - c24 = 0.
PART II.
The expression of an integral function whose zeros are given.
9. In what follows we consider a space of n dimensions, n being even and equal
to 2p. The points of this space being as before given by the n coordinates xl,..., x,
we define from these p complex variables by means of the equations,. = 21-i + ix2r, (r = 1, 2,..., p).
As it is desirable to take the various points separately we begin by supposing that
we have defined in this space an (n - 2)-fold, given in sufficiently near neighbourhood
of any point (x1(),..., xn(0)) of itself by the vanishing of an ordinary power series in the
quantities - (, 0).,.. -p(), where ~) =- X(L + ix(O) We proceed to shew that the
(n - 2)-fold can be given throughout its extent by the vanishing of a single-valued
integral function of l,.., - p (~ 15).
Such an (n - 2)-fold, given by relations involving only complex variables, may be
called a complex (n - 2)-fold; its direction cosines satisfy particular relations, as we now
prove. It is determined in sufficiently near neighbourhood of any point of itself by the
two equations arising, say, from
+ (1,..., ^)= + = O,
where u, v are real functions of the n real variables x,.., xn, which satisfy the
equations
au av au 3v
OXî_r-, =x-r' X =r - 2_r-;